Relaxation to equilibrium in controlled-not quantum networks

The approach to equilibrium of quantum mechanical systems is a topic as old as quantum mechanics itself, but has recently seen a surge of interest due to applications in quantum technologies, including, but not limited to, quantum computation and sensing. The mechanisms by which a quantum system approaches its long-time, limiting stationary state are fascinating and, sometimes, quite different from their classical counterparts. In this respect, quantum networks represent mesoscopic quantum systems of interest. In such a case, the graph encodes the elementary quantum systems (say qubits) at its vertices, while the links define the interactions between them. We study here the relaxation to equilibrium for a fully connected quantum network with controlled-not (cnot) gates representing the interaction between the constituting qubits. We give a number of results for the equilibration in these systems, including analytic estimates. The results are checked using numerical methods for systems with up to 15–16 qubits. It is emphasized in which way the size of the network controls the convergency.

Figure 1

Distance $d$ in Eq. (3) between evolving states and their corresponding asymptotic limits: We considered a fully connected network of six qubits with equally distributed weights. The (blue) solid line represents the upper bound in Eq. (3). The (green) dotted, (red) dash-dotted, and (light-blue) dashed lines correspond to initial states $|000001\rangle ,|101010\rangle ,|111111\rangle$, respectively.

Figure 2

Graph ${\mathsf{G}}_{\varphi }$ induced by graph of interaction: four-qubit oriented star. Vertices correspond to elements of computational basis. Two vertices are linked iff one of them is an image of the other under some cnot gate from the interaction graph. The list of pairs of indices are the cnot gates which contribute to weights of a given edge.

Figure 3

Fully connected graph with equally distributed weights. Dots: algebraic connectivity. Dash-dotted (green) line: bound (35).

Figure 4

Dots: algebraic connectivity. Dotted (green) line, fit in Eq. (37), with $a=0.704$ and $b=0.030$, obtained by fitting the points $N\ge 8$.

Figure 5

Same as in Fig. 4, for the circle graph. Dots: algebraic connectivity. Dotted (green) line: fit in Eq. (38), with $a=0.301$ and $b=-1.888\times {10}^{-4}$, obtained by fitting the points $N\ge 8$.

Figure 6

Algebraic connectivity of a fully connected graph vs $N$, for different noise on the probabilistic weights $p_{ij}$. Noise $\epsilon$ increases from top to bottom: Full (blue) line, noiseless; dashed (orange) line, $\epsilon =0.3$; dotted (green) line, $\epsilon =0.6$; dashed-dotted (red) line, $\epsilon =1.0$. The vertical bars are standard deviations. The fits (not shown) always yield a dependence ${\gamma }_{\varphi }=a/N+b/N^2$, with ($\epsilon =0.3$) $a=0.695,b=-0.117$; ($\epsilon =0.6$) $a=0.735,b=-1.013$; ($\epsilon =1.0$) $a=0.702,b=-1.763$.

Figure 7

Same as in Fig. 6, for a circle graph. Noise $\epsilon$ increases from top to bottom: Full (blue) line, noiseless; dashed (orange) line, $\epsilon =0.3$; dotted (green) line, $\epsilon =0.6$; dashed-dotted (red) line, $\epsilon =1.0$. The vertical bars are standard deviations.

Figure 8

A very unbalanced fully connected graph. $N(N-1)-1$ links have probabilities $\epsilon =O\left(N^{-3}\right)$, while one link has probability $O\left(1\right)$. Dots: algebraic connectivity. Dotted (green) line: fit in Eq. (40), with $a=0.629$ and $b=-3.831$, obtained by fitting the points $N\ge 8$. Dash-dotted (green) line: bound (35).